Hermitian Manifolds Research Articles

An approach to the Griffiths conjecture

The Griffiths conjecture asserts that every ample vector bundle $E$ over a compact complex manifold $S$ admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ to induce a Griffiths positive $L^2$-metric on the vector bundle $E$. This result suggests to study the relative K\ahler-Ricci flow on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ for the fibration $\mathbb{P}(E^*)\to S$. We define a flow and give arguments for the convergence.

Generalized Connections, Spinors, and Integrability of Generalized Structures on Courant Algebroids

We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined on Courant algebroids. We develop a new, self-contained, approach for the theory of Dirac generating operators on regular Courant algebroids with scalar product of neutral signature. As an application we provide a criterion for the integrability of generalized almost Hermitian structures (G, \mathcal J) and generalized almost hyper-Hermitian structures (G, \mathcal J_{1}, \mathcal J_{2}, \mathcal J_{3}) defined on a regular Courant algebroid E with scalar product of neutral signature, in terms of canonically defined differential operators on spinor bundles associated to E_{\pm} (the subbundles of E determined by the generalized metric G).

GENERAL SCHWARZ LEMMAS BETWEEN PSEUDO-HERMITIAN MANIFOLDS AND HERMITIAN MANIFOLDS

<abstract> From the viewpoint of differential geometry, Schwarz lemmas of distance-decreasing type and volume-decreasing type can be obtained by the estimates of sum functions and product functions of all eigenvalues of holomorphic maps. This paper investigates general Schwarz lemmas by estimating partial sum functions and partial product functions of the eigenvalues of generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. </abstract>

Existence and uniqueness theorems for pointwise slant immersions in complex space forms

An isometric immersion f : Mn ? ?Mm from an n-dimensional Riemannian manifold Mn into an almost Hermitian manifold ?Mm of complex dimension m is called pointwise slant if its Wirtinger angles define a function defined on Mn. In this paper we establish the Existence and Uniqueness Theorems for pointwise slant immersions of Riemannian manifolds Mn into a complex space form ?Mn(c) of constant holomorphic sectional curvature c, which extend the Existence and Uniqueness Theorems for slant immersions proved by B.-Y. Chen and L. Vrancken in 1997.

RETRACTED Warped Product Pointwise Semi-slant Submanifolds of Sasakian Manifol Retracted

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds and obtain several fundamental results, including a characterization for warped product pointwise semi-slant submanifolds of Sasakian manifolds.

Morse-type integrals on non-Kähler manifolds

We pose a conjecture about Morse-type integrals in nef (1,1) classes on compact Hermitian manifolds, and we show that it holds for semipositive classes, or when the manifold admits certain special Hermitian metrics.

Locally conformally balanced metrics on almost abelian Lie algebras

Abstract We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.

Deformations of Strong Kähler with torsion metrics

Abstract Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics {ω t } t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds {M t } t .

Some notes on metallic Kähler manifolds

The present paper deals with metallic K?hler manifolds. Firstly, we define a tensor H which can be written in terms of the (0,4)-Riemannian curvature tensor and the fundamental 2-form of a metallic K?hler manifold and study its properties and some hybrid tensors. Secondly, weobtain the conditions under which a metallic Hermitian manifold is conformal to a metallic K?hler manifold. Thirdly, we prove that the conformal recurrency of a metallic K?hler manifold implies its recurrency and also obtain the Riemannian curvature tensor form of a conformally recurrent metallic K?hler manifold with non-zero scalar curvature. Finally, we present a result related to the notion of Z recurrent form on a metallic K?hler manifold.

Fully non-linear parabolic equations on compact Hermitian manifolds

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Sz\'ekelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

The complex Monge–Ampère equation with a gradient term

We consider the complex Monge-Amp\`ere equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.

Optimal L2 extension of sections from subvarieties in weakly pseudoconvex manifolds

In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex K\"{a}hler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.

On holomorphic mappings between almost Hermitian manifolds

Our goal is to combine the techniques of Xiaokui Yang, Valentino Tosatti, and others to establish a Liouville-type result for almost complex manifolds. The transition to the non-integrable setting is delicate, so we will devote a section to discuss the key differences, and another to introduce the tools we will be using. Afterwards, we present a proof of our main theorem.

Analytically stable Higgs bundles on some non-Kähler manifolds

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kahler, we solve the Hermitian–Einstein equation on analytically stable Higgs bundles.

The continuity equation of almost Hermitian metrics

We extend the continuity equation of the Kähler metrics introduced by La Nave & Tian and the Hermitian metrics introduced by Sherman & Weinkove to the almost Hermitian metrics, and establish its interval of maximal existence. As an example, we study the continuity equation on the (locally) homogeneous manifolds in more detail.

Second order estimates for complex Hessian equations on Hermitian manifolds

<p style='text-indent:20px;'>We derive second order estimates for <inline-formula><tex-math id="M1">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>-plurisubharmonic solutions of complex Hessian equations with right hand side depending on the gradient on compact Hermitian manifolds.

On the Kähler-likeness on nearly Kähler manifolds

We introduce Kähler-like, G-Kähler-like almost Hermitian metrics. We characterize the Kähler-likeness and the G-Kähler-likeness, and show that these properties are equivalent on nearly Kähler manifolds. Furthermore, we prove that a nearly Kähler manifold with the Kähler-likeness is Kähler.

A new positivity condition for the curvature of Hermitian manifolds

In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-Kahler case. We derive a Bochner formula for closed (1, 1)-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfies our positivity condition, then any class $$\alpha \in H^{1, 1}_{BC}(X)$$ can be represented by a closed (1, 1)-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.

Invariant $${\mathcal {G}}_1$$ structures on flag manifolds

Let $${\mathbb {F}}_{\Theta }=U/K_\Theta $$ be a partial flag manifold, where $$K_\Theta $$ is the centralizer of a torus in U. We study U-invariant almost Hermitian structures on $${\mathbb {F}}_{\Theta }$$ . The classification of these structures are naturally related with the system $$R_{\mathfrak {t}}$$ of $${\mathfrak {t}}$$ -roots associated to $${\mathbb {F}}_{\Theta }$$ . We introduced the notion of connectedness by triples with zero sum in a general subset of a vector space and proved that the set of $${\mathfrak {t}}$$ -roots satisfies this property. Using this result, the invariant $${\mathcal {G}}_1$$ structures on $${\mathbb {F}}_{\Theta }$$ are completely classified.

Analytic characterization of nef and good line bundles

We give an analytic characterization of nef and good line bundles on smooth projective varieties. This supplements F. Russo's paper ``A characterization of nef and good divisors by asymptotic multiplier ideals'' (Bull. Belg. Math. Soc. Simon Stevin 2009) and extends it in the analytic direction of singular hermitian metrics.